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Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields.Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement calculus, descriptions, and equality. Succeeding chapters explore abstract set theorywith examinations of class membership as well as relations and functionscardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use.

This substantial 1953 book is a modernized presentation of the topics and approach of the Principia Mathematica of Whitehead and Russell (1910, 1912, 1913). So this would be a great companion book to that famous 3-volume work.

In this Logic for Mathematicians Dover book by John Barkley Rosser, the Whitehead/Russell theory of types is replaced by "Quine's New Foundations" (page 206), which is an unfortunate choice, but nowhere near as bad as the Whitehead/Russell theory of types!

Rosser uses the almost unreadable Peano-style dot-notation instead of modern grouping-parentheses, but does use parentheses for logical quantifiers. So the set of non-negative integers is defined like this:

x((β)::0∈β:(y).y∈β⊃y+1∈β.:⊃:.x∈β),

which I think most people would find difficult to parse! (There's actually a hat on the first "x" which I can't type in HTML.) By the way, that definition of the integers (if you do make the effort to parse it), says effectively that the integers are the intersection of all classes which contain zero and all of its successors. That's not the kind of definition which most working mathematicians would prefer to use these days.

Rosser gives an excellent presentation of the possible solutions to Russell's paradox on pages 197-207. There are many fascinating historical discussions, as well as lengthy comments on the application of logic to practical mathematics.

Pages 12-76 present a 3-axiom propositional calculus for the operators "⇒", "∧" and "¬", using modus ponens. Then there are presentations of predicate calculus with and without equality. Then classes are axiomatized. This is followed by relations and functions, cardinal numbers (111 pages: 345-455), ordinal numbers, the integers, and the axiom of choice. Unfortunately, this is all done in archaic notations in an old-fashioned style. This book has considerable historical interest, but it cannot be recommended as an introduction to modern logic.

PS. 2013-10-8.My opinion of this book has increased recently. In particular, I am very happy with Rosser's treatment of predicate calculus. I am referring here particularly to "Rule C" on pages 126-149, which according to Margaris, First Order Mathematical Logic, page 191, was introduced by Rosser, who shows that any theorem which can be proved with Rule C can be proved without Rule C. This rule is a "choice rule", which is how every real mathematician uses existential quantifiers in proofs. And this shows the enormous strength of this book by Rosser. He presents logic in a way which is consistent with how mathematicians do mathematics. Most logic books before Rosser were using an equivalent of Rule C, called "existential quantifier elimination", or EE for short. But after Rosser, this rule has been omitted by most logic textbooks. So this book by Rosser seems to mark the point in history that predicate calculus treatments abandoned the natural way in which mathematicians really work.

And just one other point about the dot-notation which Rosser uses. The best way to interpret it is to snip the expression where the largest set of dots occurs, and put parentheses around both halves. Then do this recursively until you have a parenthesized expression.

Another really good thing about the Rosser book is that it gives applications to the foundations of mathematics in a way which is useful to mathematicians, following the topic sequence of the Whitehead and Russell 3-volume Principia Mathematica. Most other logic textbooks roam off into proof theory and model theory and extreme abstractions which have little relevance to mathematicians. So if you want logic which is useful for doing mathematics, this might be the right book for you.

While I think there are better books on first order logic (Quine's, for instance), this one is still quite good. What makes this book truly remarkable, though--unique, even--is its coverage of set theory. It develops the subject matter of a good elementary set theory text, but does it within the framework of Quine's New Foundations (NF) set theory. NF is the chief alternative set theory to Zermelo-Fraenkel and related systems, and no other text that I'm aware of develops NF's theory of cardinals and ordinals in anywhere near as much detail. For those wishing to study NF, or to see mathematics implemented more fully in radically different set theory, this book is an indispensible resource.

I am just getting started, but I must say for someone that has never truly peeled the onion but rather has taken a number of half-baked stabs at the subject from the need for analysis proofs perspective, Rosser has the unusual knack of being able to oscillate the reader between a pleasant bewilderment to enlightenment often times over the course of a single sentence.